Skip to main content

Geometries of the projective matrix space

  • 341 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1165)

Abstract

In our paper on matrix Möbius transformations (Comm. Algebra 9 (19) (1981), 1913–1968) we introduced the one-dimensional left-projective space over the complex n×n matrices P = P1(Mn(ℂ)). For n = 1 this space is the projective complex line P1(ℂ) and so is homeomorphic to the Riemann sphere. We studied the topology of P and the projective mappings of P onto itself. Here we present some generalizations of certain parts of the Euclidean, the spherical and the non-Euclidean geometry from the scalar to the multidimensional case.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BUSEMANN, H, Convex surfaces, Interscience, New York 1958.

    MATH  Google Scholar 

  2. CARATHEODORY, C., Theory of functions of a complex variable, Vol.1, Chelsea, New York 1954.

    MATH  Google Scholar 

  3. GANTMACHER, F.R., The theory of matrices, Vol.1, Chelsea, New York 1959.

    MATH  Google Scholar 

  4. HUA, L.K., Geometries of matrices III. Fundamental theorems in the geometries of symmetric matrices, Trans. Amer. Math. Soc. 61 (1947), 229–255.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. KELLEY, J.L., General topology, Van Nostrand, Princeton, N.J. 1955.

    MATH  Google Scholar 

  6. LANCASTER, P., Theory of matrices, Academic Press, New York 1969.

    MATH  Google Scholar 

  7. LONDON, D., An inequality for the spectral norm of certain matrices, Linear and Multilinear Algebra, 14(1983), 37–44.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. MARSHALL, A.W. and I. OLKIN, Inequalities: theory of Majorization and the applications, Academic Press, New York 1979.

    MATH  Google Scholar 

  9. POTAPOV, V.P., The multiplicative structure of J-contractive matrix functions, Amer. Math. Soc. Translations, Ser. II 15 (1960), 131–243.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. SCHWARZ, B. and A. ZAKS, Matrix Möbius transformations, Comm. Algebra (9) (19) (1981), 1913–1968.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. SIEGEL, C.L., Symplectic geometry, Amer. J. Math. 65 (1943), 1–86.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Schwarz, B., Zaks, A. (1985). Geometries of the projective matrix space. In: Ławrynowicz, J. (eds) Seminar on Deformations. Lecture Notes in Mathematics, vol 1165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076159

Download citation

  • DOI: https://doi.org/10.1007/BFb0076159

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16050-2

  • Online ISBN: 978-3-540-39734-2

  • eBook Packages: Springer Book Archive